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The curves x^2 + y^2 + 4x + 6y = 12 and 5y = 4x + 13 are drawn in the coordinate plane.  How many times do they intersect?

 Dec 17, 2019
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x^2 + y^2 + 4x + 6y = 12 and 5y = 4x + 13 

 

Rearrange as the second equation as

 

y  = [ 4x + 13 ] / 5

 

Sub this into the second equation

 

x^2  +  ( [ 4x + 13]/ 5)^2 + 4x + 6 ( [ 4x + 13' / 5)  =  12    simplify

 

x^2  +  [ 16x^2 + 104x + 169 ] / 25  + 4x  + [ 24x + 78 ] / 5   =  12     multiply through by 25

 

25x^2 + 16x^2 + 104x + 169  + 100x  + 120x + 390  =  300

 

41x^2 + 324x + 259  =  0        factor as

 

(41x  + 37)  ( x + 7)   =  0

 

Set each factor to 0  and   solve for x   and we get that

 

x = -37/41        and   x   = -7

 

This indicates that there are two  intersection points

 

 

cool cool cool

 Dec 17, 2019

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